A New Logical Framework for Deductive Planning *

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1 A New Logical Framework for Deductive Planning * Werner Stephan and Susanne Biundo German Research Center for Artificial Intelligence (DFKI) Stuhlsatzenhausweg 3, Saarbrucken <lastname>@dfki.uni-sb.cie Abstract In this paper we present a logical framework for defining consistent axiomatizations of planning domains. A language to define basic actions and structured plans is embedded in a logic. This allows general properties of a whole planning scenario to be proved as well as plans to be formed deductively. In particular, frame assertions and domain constraints as invariants of the basic actions can be formulated and proved Even for complex plans most frame assertions are obtained by purely syntactic analysis In such cases the formal proof can be generated in a uniform way. The formalism we introduce is especially useful when treating recursive plans A tactical theorem prover, the Karlsruhe Interactive Verifier KIV is used to implement this logical framework 1 Introduction In this paper we present a logical framework for defining consistent axiomatizations of planning domains. An effective mechanism for defining the basic operations in a constructive way is embedded in a logic that allows properties of state spaces given by these actions to be proved. This not only includes the deductive generation of plans; reasoning about the complete scenario is also supported by this approach Our work follows the "Plans are Programs" paradigm. This approach is not new; in deductive planning, in particular, it has already been discussed by several authors (cf. [Green, 1969; Rosenschein, 1981; Kautz, 1982; Bibel, 1986; Manna and Waldinger, 1987; Biundo et al., 1992]). However, their contributions concentrated mainly on aspects of common control structures like sequential composition, conditional branching, and sometimes recursion. Less attention has been paid to the question of data structures Our approach is based on the idea of treating relations used in the axiomatizations of planning scenarios in the same way data with an algebraic structure are treated in ordinary programming lan- *This work was partly supported by the German Ministry for Research and Technology (BMFT) under contract ITW guages. In planning, situations are usually described by a set of relations between certain objects (blocks, rooms, robots etc.). These relations are flexible in the sense that they may change from one situation to another. The fact that these changes are local to a small section of the entire situation is not reflected in the basic semantic concepts underlying most formalisms for deductive planning. So, for example, the blocks world operation unstack(a,b), [Genesereth and Nilsson, 1987], changes only the on-relation between a and 6, the clear-property of 6 and the table-property of a If the pile of blocks is just a part of some room, which in turn is just one constituent of a larger scene, the unstack-operation exhibits very local behavior. A straightforward concept is to consider these relations as objects, like elements of (abstract) data types in ordinary programming languages In the theory of abstract data types most often one only considers algebraic structures where all elements are (freely) generated by so-called constructors. In our case, restricting ourselves to finite relations, an appropriate set of Constructors" and "selectors" can easily be devised as well Starting with the empty relation all finite n-ary relations can be generated by successive applications of an add- operation that adds an n-tuple to a given relation As is the case with freely generated data types, we have to supply a corresponding delete-operation in order to compute with this data type It seems reasonable to take these two operations as the basis of a planning language designed to compute changes of relational structures. In our approach relations between unstructured objects correspond to data objects and are therefore considered to be finite. This seems to be a realistic assumption in most applications. In the context of recursive plans the finiteness of relations and its reflection by axioms becomes essential for termination proofs. This approach, whereby one considers a fixed state space generated by add- (and delete') operations, underlies STRIPS-like [Fikes and Nilsson, 1971] planners. These systems also demand a complete description of the operations used in the planning process. In particular, they allow an efficient treatment of the frame problem [McCarthy and Hayes, 1969] since it is explicitly denoted which parts of a situation are changed. Our approach lends formal semantics to the basic concepts 32 Automated Reasoning

2 underlying the STRIPS approach and, while retaining its effectiveness, removes most of its limitations by providing a more general mechanism for defining operations and by embedding this mechanism in a logical framework. This not only extends the STRIPS approach and makes it suitable for deductive planning but also allows whole planning environments to be set up in a provably consistent way. Based on elementary add- and delete-operations with fixed semantics, we will begin with the definition of basic actions, and then use efficient control structures to build up more complex plans. Apart from the usual ones, these control structures include a new non-deterministic choose construct, necessary to select objects in a nondeterministic way. Domain constraints are formulated as tnvariants of the basic actions and can be proved from their basic definitions, thus guaranteeing consistency of the whole planning environment. Due to the fixed semantics of the elementary operations, as well as the control structures, "side effects" can be excluded in many cases, by a purely syntactical inspection of plans, thereby providing an efficient treatment of the frame problem. We will restrict ourselves to a formalization within a variant of Dynamic Logic (DL), although many of the basic ideas could be used in the context of other programming logics as well (e.g., [Salwicki, 1977; Harel, 1979; Manna and Pnueli, 1991; Kroger, 1987]). Not only does DL seem to be expressive enough for most applications in planning; this choice also allows an easy implementation of our formalism in an existing deductive system, the Karlsruhe Interactive Verifier (KIV) [Heisel et a/., 1990]. This system can be used as a logic-based shell for setting up planning environments. This includes the definition of the basic actions, the proof of invariants and additional lemmmata, and on top of that the implementation of various planning strategies The paper is organized in the following way: section 2 introduces the semantic background of our theory. In section 3, we define the logic, including syntax and semantics of the planning language. Section 4 shows how state invariants can be derived from the basic operator definitions; it also shows how the abstract operator descriptions, already seen in other planning formalisms, can be obtained, and serve as the basis for deductive planning Section 5 is devoted to the frame problem In section 6 we discuss some aspects of the implementation within the KIV system. Section 7 refers to related work and finally, we conclude with some remarks in section 8. Stephen and Biundo 33

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6 They define basic actions as atomic constituents of their planning language that are axiomatized freely by describing their preconditions and effects, respectively. Our approach goes beyond this by providing a STRIPSlike way of defining basic actions and setting up consistent planning scenarios on top of that. The logical formalism is extended to reason about these basic actions as well as about composite plans built out of them. In both cases this includes recursive definitions. With the work of Pednault [Pednault, 1986; Pednault, 1989] the approach we presented in this paper shares the idea of describing basic actions in a STRIPS-like manner, that is, by giving add and delete lists for relations. Moreover, both approaches embed these descriptions into a logical formalism that can be used to reason about plans. We begin with the observation that the appropriate semantical background for integrating this STRIPS approach into deductive planning are models based on finitely generated relations. While ADL uses a fixed form of conditional add and delete lists our approach allows to program basic operations in a carefully chosen programming language that covers ADL schemata in a straightforward way. In our approach it is easy to add non-determinism and also in the deterministic case we can do without auxiliary relations which seem to be necessary in ADL to describe more complicated actions. In our setting we start out with the definition of basic actions. The defining programs always have a precise meaning in the underlying semantical structures. From these definitions which in addition are independent of each other we then prove domain constraints, derived descriptions, and frame assertions. These issues are not addressed in the ADL work. In addition, we have outlined a method to generate certain frame assertions even for composite plans by a purely textual analysis. Although many ideas presented above are independent of the logical basis we want to stress that the ability to reason about the structure of (possibly) recursive definitions (programs) is essential in this context. As already mentioned above, our formalism is based on the STRIPS ideas that have been given formal semantics by Lifschitz [Lifschitz, 1986]. We feel that our formalism in some sense "proceduralizes" Lifschitz's approach and extends it in some way, e.g. as far as the treatment of negative effects etc. is concerned. However, investigating this relationship in more detail goes beyond the scope of this paper. Separate work will be devoted to that issue. 8 Conclusion Combining characteristic features of conventional planning with techniques borrowed from programming logics, we have introduced a new theory of action based on a special variant of Dynamic Logic. Plans may be constructed using rich control structures including recursion, non-deterministic branching, and a special choose construct. In our approach, we start out by the definition of basic actions and are then able to prove properties about the state space generated by these actions. This includes frame assertions and domain constraints. In this way, we prevent our planning environment from running into inconsistencies. These are possible in other Stephan and Biundo 37

7 systems where frame assertions and domain constraints are considered to be axioms. An efficient treatment of the frame problem is provided by a method to generate most frame assertions non-deductively (with the possibility of a uniform formal proof within the system). Our theory of action clearly is not restricted to blocks-worldtype planning domains. One could equally well define a theory for an intelligent help system context where the planning domain is a command language environment. There the ability of reasoning about recursive plans is essential. Implementing our logical framework in the KIV system provides the basis not only for a deductive planning system, but also for a complete deductive planning environment, i.e., a system that also assists a user in developing a consistent axiomatization of his planning domain. Furthermore, this notion of environment can be extended by implementing tactics for temporal projection, plan validation and other reasoning methods. Further work is devoted to the automated generation of recursive plans and an extension of the logical framework to parallelism. References [Bibel et a/., 1989] W. Bibel, L. Farinas del Oerro, B. Fronhofer, and A. Herzig. Plan Generation by Linear Proofs: On Semantics. In GWA189: Curman Workshop on Artificial Intelligence, pages Springer LNCS 216, [Bibel, 1986] W. Bibel. A Deductive Solution for Plan generation. New Generation Computing, 4: , [Biundo et al, 1992] S. Bitindo, D. Dengler, and J. Kohler. Deductive Planning and Plan Reuse in a Command Language Environment. In Proceedings of the 10th European Conference on Artificial Intelligence, pages , [Dijkstra, 1976] E.W. Dijkstra. A Discipline of Programming. Prentice Hall, London, [Fikes and Nilsson, 1971] RE. Fikes and N..I Nilsson. STRIPS: A New Approach to the Application of Theorem Proving to Problem Solving. Artificial Intelligence, 2: , [Genesereth and Nilsson, 1987] MR. Genesereth and N.J. Nilsson. 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